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Sustainable Energy Technologies and Assessments 6 (2014) 86–92
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Sustainable Energy Technologies and Assessments
journal homepage: www.elsevier .com/locate /seta
Technical Note
Hourly solar radiation on inclined surfaces
http://dx.doi.org/10.1016/j.seta.2014.01.0072213-1388/� 2014 Published by Elsevier Ltd.
⇑ Corresponding author. Tel: +91 9918676947.E-mail address: [email protected] (C.K. Pandey).
C.K. Pandey a,⇑, A.K. Katiyar b
a Department of Applied Sciences & Humanities, R R Institute of Modern Technology, Bakshi Ka Talab, Sitapur Road, Lucknow, Indiab Institute of Engineering & Technology, Gautam BudhaTechnical University, Lucknow, India
a r t i c l e i n f o
Article history:Received 22 March 2013Revised 27 December 2013Accepted 24 January 2014
Keywords:Solar radiationHorizontal surfaceInclined surfaceStatistical error
a b s t r a c t
The present study investigates the variation of hourly global solar radiation for inclined surfaces. We havemeasured the data for horizontal surface and different south-facing (c = 0�) inclined surfaces at Lucknow(Latitude 26.75�, Longitude 80.85�), Uttar Pradesh, India. These inclined surfaces data have been com-pared with the estimated tilted results from measured horizontal data using circumsolar model, the iso-tropic model suggested by Liu and Jordan [1], and the two anisotropic models by Klucher [3] and Hay [2].Statistical error tests are also performed for the critical analysis of the models. The comparative study ofthe model validates the performance of isotropic model for all inclined surfaces.
� 2014 Published by Elsevier Ltd.
1. Introduction
Having the definite values of latitude/longitude of Indian cities,the energy systems work more efficiently if their receiving surfacesare accordingly inclined. Hence, it is very important to have theknowledge of solar radiation on inclined surfaces. Although solarradiation, including the global and diffuse components, is mea-sured on horizontal surfaces at many locations, measurement onan inclined surface is very rare. Due to the lack of measured tiltedsurface solar radiation data, models are employed to estimate theradiation incident on a collector surface from the measured hori-zontal radiation. Over the years, different mathematical modelshave been developed by Liu and Jordan [1], Hay [2], Klucher [3],Heywood [4], Klein [5], and others, to calculate total radiation ontilted surfaces from measurements of horizontal surface. Gopina-than [6] and Nijmeh and Mamlook [7] have compared one isotropicand one anisotropic model for estimating daily/hourly total radia-tion on tilted surfaces in Lesotho and Jordan, respectively, and sug-gested that both the models are equally accurate for stations inLesotho [8,13]. Recently, Kamali et al. [8] and Loutzenhiser et al.[9] used the various models to estimate the solar radiation overtilted plane in Iran and Switzerland respectively. A few attemptsto develop knowledge for tilted surfaces have been found in the lit-eratures [11–13].
The prime objective of the present study is (i) the analysis of themonthly mean hourly total radiation on an inclined south facing
(c = 0�) surface with the variation of 15�–60� inclination anglesand (ii) to compare, statistically, widely used models for estimatingsolar radiation on various inclined surfaces for their applicability inthe plane area of northern India.
2. Methodology
In this section, we have described about the site where mea-surement carried out and estimation methodology.
2.1 Site description
From a solar system point of view, it is of interest to comparethe ability of models to estimate the energy on a tilted surface.Therefore, we have measured solar radiation (global and diffuse)on horizontal surface and tilted south facing (c = 0�) surfaces in-clined at 15�, 30�, 45�, and 60�. The measurements using a preci-sion Pyranometer, having a calibration factor 5.37 mv/cal/cm2/min and the shading ring was made over a period of one year fromJanuary to December 2007 at the solar energy laboratory of theInstitute of Engineering and Technology, Lucknow (Latitude26.75�, Longitude 80.85�, Altitude 120 meter above sea level),Uttar-Pradesh, India [12–13]. The Shading ring blocks the directincident radiation on the dome of the Pyranometer and allows onlydiffuse radiation.
2.2 Estimation methodology
The monthly mean hourly total radiation (It) on a tilted surfaceis made up of the beam radiation (Ib), sky diffuse radiation (Is) and
Table 1Statistical test results (W/m2) for calculated and measured solar radiation data on a tilted surface.
Slope Month Circumsolar Isotropic Klucher Hay
RMSE MBE RMSE MBE RMSE MBE RMSE MBE
15� February 166.19 149.17 31.16 17.55 79.08 67.78 105.88 81.29April 33.36 �19.02 33.56 �26.99 27.11 �9.89 32.92 �23.56June 69.27 �53.49 56.1 �44.36 48.85 �31.38 60.68 �47.78September 129.95 95.15 116.39 76.99 134.37 99.79 121.84 84.51December 57.64 51.88 5.19 2.85 36.35 30.67 19.87 17.98
30� February 91.48 60.37 70.28 �9.61 86.78 29.47 72.37 18.78April 55.96 �31.45 50.87 �40.29 42.67 �22.67 52.2 �36.33June 93.29 �85.72 64.43 �60.13 55.92 �47.89 74.99 �70.19September 29.11 15.91 22.1 �12.68 22.83 11.8 20.92 �0.82December 91.31 75.73 24.11 �14.52 31.87 23.63 23.25 58.08
45� February 109.07 104.86 28.6 14.36 66.47 57.73 59.93 50.94April 95.68 �88.97 93.16 �91.34 75.73 �72.06 93.35 �89.97June 157.2 �153.07 107.02 �104.79 93.28 �89.88 126.53 �124.28September 28.19 4.03 34.79 �26.52 23.57 �0.52 28.05 �13.74December 109.64 102.66 48.75 �36.12 22.2 9.03 26.14 0.94
60� February 101.67 58.58 18.84 �0.69 50.37 43.73 46.94 39.24April 121.45 �117.37 109.04 �106.31 87.24 �83.46 113.37 �110.29June 225.7 �222.69 149.94 �147.07 128.92 �125.67 180.41 �178.09September 33.62 �29 57.43 �52.94 29.83 �24.89 46.71 �42.77December 112.28 78.47 76.05 �60.22 27.69 �12.72 39.4 �17.88
6 8 10 12 14 16 18-100
0
100
200
300
400
500
600
700
800
900
1000
Beta = 150
Mon
thly
mea
n ho
urly
tilte
d gl
obal
radi
atio
n (W
/m2 )
Time
Circumsolar Isotropic Klucher Hay Measured
(a)
6 8 10 12 14 16 18-100
0
100
200
300
400
500
600
700
800
900
1000
Beta = 450
Mon
thly
mea
n ho
urly
tilte
d gl
obal
radi
atio
n (W
/m2 )
Time
Circumsolar Isotropic Klucher Hay Measured
(c)
6 8 10 12 14 16 18-100
0
100
200
300
400
500
600
700
800
900
1000
Beta = 300
Mon
thly
mea
n ho
urly
tilte
d gl
obal
radi
atio
n (W
/m2 )
Time
Circumsolar Isotropic Klucher Hay Measured
(b)
6 8 10 12 14 16 18-100
0
100
200
300
400
500
600
700
800
900
1000
Beta = 600
Mon
thly
mea
n ho
urly
tilte
d gl
obal
rad
iatio
n (W
/m2 )
Time
Circumsolar Isotropic Klucher Hay Measured
(d)
Fig. 1. Measured and estimated global solar radiation received on inclined surfaces for the month of February, Lucknow, India.
C.K. Pandey, A.K. Katiyar / Sustainable Energy Technologies and Assessments 6 (2014) 86–92 87
reflected ground radiation (Ir). Thus, for a surface tilted with re-spect to the horizontal, the incident total radiation is
It ¼ Ib þ Is þ Ir ð1Þ
The hourly beam radiation received on the tilted surface can beexpressed as:
Ib ¼ ðI � IdÞrb ð2Þ
6 8 10 12 14 16 18-100
0
100
200
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400
500
600
700
800
900
1000
Beta = 150
Mon
thly
mea
n ho
urly
tilte
d gl
obal
rad
iatio
n (W
/m2 )
Time
Circumsolar Isotropic Klucher Hay Measured
(a)
6 8 10 12 14 16 18-100
0
100
200
300
400
500
600
700
800
900
1000
Beta = 450
Mon
thly
mea
n ho
urly
tilte
d gl
obal
rad
iatio
n (W
/m2 )
Time
Circumsolar Isotropic Klucher Hay Measured
(c)
6 8 10 12 14 16 18-100
0
100
200
300
400
500
600
700
800
900
1000
Beta = 300
Mon
thly
mea
n ho
urly
tilte
d gl
obal
rad
iatio
n (W
/m2 )
Time
Circumsolar Isotropic Klucher Hay Measured
(b)
6 8 10 12 14 16 18-100
0
100
200
300
400
500
600
700
800
900
1000
Beta = 600
Mon
thly
mea
n ho
urly
tilte
d gl
obal
rad
iatio
n (W
/m2 )
Time
Circumsolar Isotropic Klucher Hay Measured
(d)
Fig. 2. Measured and estimated global solar radiation received on inclined surfaces for the month of April, Lucknow, India.
Model Year Rd=
Circumsolar 1983 rb ¼ sin d sinðU�bÞþcos d cosðU�bÞ cos xsin d sin Uþcos d cos U cos x ,
whered = 23.45� sin[(284+n)360/365]x = cos�1(-tanUtand)
Liu and Jordan 1962 12 ð1þ cos bÞ
Hay 1979 I�IdIo
rb þ 12 ð1þ cos bÞ 1� I�Id
Io
h in o
Klucher 1979 12 ð1þ cos bÞ½1þ F sin3ðb=2Þ�ð1þ F cos2 h sin3 hzÞ;, whereF = 1-(Id/I)2
cos h = sin d sin (U-b) +cos d cos (U-b) cos xsin hz ¼
ffiffiffi1p� ðsin d sin Uþ
cos d cos U cos xÞ2
88 C.K. Pandey, A.K. Katiyar / Sustainable Energy Technologies and Assessments 6 (2014) 86–92
where I and Id are monthly mean hourly total and diffuse radiation,respectively, measured on horizontal surface and
rb ¼sin d sinðU� bÞ þ cos d cosðU� bÞ cos x
sin d sin Uþ cos d cos U cos xð3Þ
U is the latitude (degree) and b the surface slope from the hor-izontal (degree). The declination for the day d (degree) and sun risehour angle x for the tilted surface are expressed as [12]:
d ¼ 23:45� sin ð284þ nÞ360=365½ � and ð4Þ
x ¼ cos�1ð� tan U tan dÞ ð5Þ
Is and Ir depend on diffuse and total radiation on horizontal surface,respectively and are expressed as:
Is ¼ Idrd ð6Þ
Ir ¼ Irr ð7Þ
where rd and rr are the ratios of hourly radiation on a tilted surfacefor diffuse and for reflected radiation to that of a horizontal surfacerespectively and Id and I are hourly diffuse and total radiation on ahorizontal surface. The models chosen for study of hourly sky dif-fuse radiation are circumsolar model [10] and the isotropic models
of Liu & Jordan [1], and the anisotropic models of Hay [2], andKlucher [3]. The information about models and mathematical rela-tionships has been taken from Pandey and Katiyar [13] referred inTable 1. We have also presented the highlights of the selected meth-od in this paper.
6 8 10 12 14 16 18-100
0
100
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1000
Beta = 150
Mon
thly
mea
n ho
urly
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d gl
obal
radi
atio
n (W
/m2)
Time
Circumsolar Isotropic Klucher Hay Measured
(a)
6 8 10 12 14 16 18-100
0
100
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500
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700
800
900
1000
Beta = 450
Mon
thly
mea
n ho
urly
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d gl
obal
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atio
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/m2 )
Time
Circumsolar Isotropic Klucher Hay Measured
(c)
6 8 10 12 14 16 18-100
0
100
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1000
Beta = 300
Mon
thly
mea
n ho
urly
tilte
d gl
obal
radi
atio
n (W
/m2 )
Time
Circumsolar Isotropic Klucher Hay Measured
(b)
6 8 10 12 14 16 18-100
0
100
200
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400
500
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900
1000
Beta = 600
Mon
thly
mea
n ho
urly
tilte
d gl
obal
radi
atio
n (W
/m2 )
Time
Circumsolar Isotropic Klucher Hay Measured
(d)
Fig. 3. Measured and estimated global solar radiation received on inclined surfaces for the month of June, Lucknow, India.
6 8 10 12 14 16 18-100
0
100
200
300
400
500
600
700
800
900
1000
Beta = 150
Mon
thly
mea
n ho
urly
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d gl
obal
radi
atio
n (W
/m2 )
Time
Circumsolar Isotropic Klucher Hay Measured
(a)
6 8 10 12 14 16 18-100
0
100
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400
500
600
700
800
900
1000
Beta = 450
Mon
thly
mea
n ho
urly
tilte
d gl
obal
radi
atio
n (W
/m2 )
Time
Circumsolar Isotropic Klucher Hay Measured
(c)
6 8 10 12 14 16 18-100
0
100
200
300
400
500
600
700
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900
1000
Beta = 300
Mon
thly
mea
n ho
urly
tilte
d gl
obal
radi
atio
n (W
/m2 )
Time
Circumsolar Isotropic Klucher Hay Measured
(b)
6 8 10 12 14 16 18-100
0
100
200
300
400
500
600
700
800
900
1000
Beta = 600
Mon
thly
mea
n ho
urly
tilte
d gl
obal
radi
atio
n (W
/m2 )
Time
Circumsolar Isotropic Klucher Hay Measured
(d)
Fig. 4. Measured and estimated global solar radiation received on inclined surfaces for the month of September, Lucknow, India.
C.K. Pandey, A.K. Katiyar / Sustainable Energy Technologies and Assessments 6 (2014) 86–92 89
6 8 10 12 14 16 18-100
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Beta = 150
Mon
thly
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n ho
urly
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d gl
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iatio
n (W
/m2 )
Time
Circumsolar Isotropic Klucher Hay Measured
(a)
6 8 10 12 14 16 18-100
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Beta = 450
Mon
thly
mea
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d gl
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iatio
n (W
/m2 )
Time
Circumsolar Isotropic Klucher Hay Measured
(c)
6 8 10 12 14 16 18-100
0
100
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Beta = 300
Mon
thly
mea
n ho
urly
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d gl
obal
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iatio
n (W
/m2 )
Time
Circumsolar Isotropic Klucher Hay Measured
(b)
6 8 10 12 14 16 18-100
0
100
200
300
400
500
600
700
800
900
1000
Beta = 600
Mon
thly
mea
n ho
urly
tilte
d gl
obal
rad
iatio
n (W
/m2 )
Time
Circumsolar Isotropic Klucher Hay Measured
(d)
Fig. 5. Measured and estimated global solar radiation received on inclined surfaces for the month of December, Lucknow, India.
90 C.K. Pandey, A.K. Katiyar / Sustainable Energy Technologies and Assessments 6 (2014) 86–92
3. Results and discussion
This research paper endeavors to compare the global solar radi-ation on inclined south facing (c = 0�) surfaces, computed from themeasured horizontal data using the four models (circumsolar, iso-tropic and two anisotropic model of hay and klucher) discussedearlier, with the measured data for Lucknow, India. The groundreflectance (albedo) is considered as 0.2. Observing the variationin slope and different seasonal and atmospheric conditions, the re-search has been designed for an entire year with the variation ofinclination angle from 15� to 60�. Due to insufficiency of space inpresenting the paper, the comparison is shown only for February,April, June, September and December months through Figs. 1–5at inclination angles 15�, 30�, 45�, and 60�.
It is clear from these figures that all the considered models givesimilar results and the same order of accuracy as the estimateddata at all inclination angles for entire year except for a slight var-iation at high inclination angles (45� and 60�). The models agreequite well with each other. A close agreement between the mea-sured and the estimated data for isotropic models is also observedfor winter months at lower inclined surfaces (15� and 30�). Analy-sis of the entire data shows that the solar radiation increases withthe increase of inclination for the winter months (October–Febru-ary). In the summer months (March–September), maximum radia-tion received at b = U. Figs. 1–5 shows that the difference between
the measured and estimated data are greater for high inclinationsduring summer months when radiation is at a peak. Statistical er-ror tests were also performed to test the accuracy of the modelsand are presented in Table 1.
The RMSE and MBE values in Table 1 show that the isotropicand anisotropic models of Hay [2] and Klucher [3] perform wellover the Circumsolar model for Indian climatic conditions. In gen-eral, the isotropic model predicts more accurate results than theHay, Klucher or Circumsolar models. A low value of RMSE andMBE shows the best performance of the isotropic model to predictthe data on tilted surfaces.
We have also compared the measured data for the months Feb-ruary, April, June, September and December on various inclinationangles (from 0� to 60�) through Fig. 6.
Fig. 6 clearly reflects that normally for entire year the maximumradiation is collected with south-facing surface tilt equal to the lat-itude (�30�). However in summer it is obtained at lower inclina-tion of 15� and at higher inclination of 45� in winter.
4. Conclusion
The hourly solar radiation on the various inclined surfaces atintervals of 15� in Lucknow was investigated by using circumso-lar, isotropic, and two anisotropic models of Hay and Klucher.
9.0 10.5 12.0 13.5 15.0 16.5 18.00
100
200
300
400
500
600
700
800
900
1000
FebruraryG
loba
l sol
ar ra
diat
ion
(W/m
2 )
Time (Hours)
Horizontal Inclined150
inclined300
Inclined450
Inclined600
9.0 10.5 12.0 13.5 15.0 16.5 18.00
100
200
300
400
500
600
700
800
900
1000
June
Glo
bal s
olar
radi
atio
n (W
/m2 )
Time (Hours)
Horizontal Inclined150
inclined300
Inclined450
Inclined600
9.0 10.5 12.0 13.5 15.0 16.5 18.00
100
200
300
400
500
600
700
800
900
1000
April
Glo
bal s
olar
radi
atio
n (W
/m2 )
Time (Hours)
Horizontal Inclined150
inclined300
Inclined450
Inclined600
9.0 10.5 12.0 13.5 15.0 16.5 18.00
100
200
300
400
500
600
700
800
900
1000
September
Glo
bal s
olar
radi
atio
n (W
/m2 )
Time (Hours)
Horizontal Inclined150
inclined300
Inclined450
Inclined600
9.0 10.5 12.0 13.5 15.0 16.5 18.00
100
200
300
400
500
600
700
800
900
1000
December
Glo
bal s
olar
radi
atio
n (W
/m2 )
Time (Hours)
Horizontal Inclined150
inclined300
Inclined450
Inclined600
Fig. 6. Comparison of the measured data of horizontal and various tilt angles (from 15� to 60�) for the month of February, April, June, September and December, Lucknow,India.
C.K. Pandey, A.K. Katiyar / Sustainable Energy Technologies and Assessments 6 (2014) 86–92 91
From the analysis, it can be concluded that (1) the maximum opti-mum solar radiation round-year received at an inclination angleequal to the latitude of the place. (2) The isotropic model by Liuand Jordan supercedes others due to its simplicity for estimatingmonthly mean hourly global solar radiation on tilted surfaces.(3) Due to the systematic analysis, the carried out research workcan serve as a useful reference for future solar energy applicationsin India.
References
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[4] Heywood H. The computation of solar energy intensities, Pt. 2, Solar radiationon inclined surfaces, Solar Energy Conf. Phoenix, AZ; 1965.
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